The MHD equations model the electrically conducting fluids in the presence of a magnetic field such as plasmas, liquid metals and electrolytes . The MHD equations play an essential role in the study of many phenomena in physics, especially, in geophysics and astrophysics. Besides their wide physical applicability, the MHD equations are also of great interest in mathematics. This talk explainsc some of the recent progress on the global well-posedness and the stability problems on the MHD equations with only partial or fractional dissipation. Our talk will be self-contained and is accessible to graduate students.